As is known, when an electrical field E penetrates inside an optically linear material, it induces inside the material a polarization P, which depends upon the electrical field E and can be expressed as:P=χ(1)E  (1)where χ(1) is a tensor of rank two, referred to as linear susceptivity.
In the case where the material is optically nonlinear, the polarization P depends upon powers of the electrical field E according to the following equation:P=χ(1)E+χ(2)EE+χ(3)EEE+  (2)where the coefficients χ(2),χ(3) etc. are defined as nonlinear susceptivity of the second order, third order, etc., and are tensors respectively of rank three, four, etc.; in other words, an n-order susceptivity χ(n) is a tensor of rank n+1.
As is known, a particular importance is assumed by the second-order nonlinear susceptivity χ(2), which is a characteristic of the materials that have a non-centrosymmetrical crystalline structure inherent in the so-called phenomenon of second-harmonic generation (SHG).
In detail, in the case where the electrical field E is oscillating with a pulsation ω, and can consequently be expressed as E=E0 cos(ωt−kz), it is possible to show that, if the higher-order terms are neglected, the polarization P is given by the following equation:P=χ(1)E0 cos(ωt−kz)+½χ(2)E02[1+cos(2ωt−2kz)]  (3)where evident is the presence of a component equal to ½χ(2)E02 cos(2ωt−2kz), which hence oscillates at a pulsation 2ω that is twice the pulsation ω of the electrical field E and is responsible for the generation of a second harmonic, i.e., of the generation of an electrical field having pulsation 2ω that is twice the pulsation ω of the electrical field E.
For the purposes of characterization of the materials, and in particular for the purposes of determination of the optical characteristics of a material, it is consequently of particular importance to determine the second-order nonlinear susceptivity χ(2), i.e., to determine the twenty-seven elements of the corresponding tensor.
From a physical standpoint, the determination of the aforementioned twenty-seven elements amounts to the determination of a subset of mutually independent elements, the remaining elements depending upon said mutually independent elements.
In detail, designating by χijk(2) (with indices i, j, k=1, 2, 3) the elements of the second-order nonlinear susceptivity tensor χ(2), it may be shown that, in the conditions where the so-called intrinsic permutation symmetry applies, i.e., assuming a stationary-state regime, the elements χijk(2) do not vary with respect to permutations of the index j and of the index k, with the consequence that the relation χijk(2)=χikj(2) applies. Consequently, on account of the intrinsic permutation symmetry, the number of independent elements χijk(2) drops from twenty-seven to eighteen.
Recalling that the nonlinear optical properties of the materials are generally expressed in terms of the so-called second-order nonlinear optical tensor {tilde over (d)}, the elements dijk, of which depend upon corresponding elements χijk(2) of the second-order nonlinear susceptivity χ(2) according to the equation dijk=0.5*χijk(2), the aforementioned second-order nonlinear optical tensor d may be expressed in the following contracted form:
                              d          ~                =                  (                                                                      d                  11                                                                              d                  12                                                                              d                  13                                                                              d                  14                                                                              d                  15                                                                              d                  16                                                                                                      d                  21                                                                              d                  22                                                                              d                  23                                                                              d                  24                                                                              d                  25                                                                              d                  26                                                                                                      d                  31                                                                              d                  32                                                                              d                  33                                                                              d                  34                                                                              d                  35                                                                              d                  36                                                              )                                    (        4        )            
The form of the second-order nonlinear optical tensor {tilde over (d)} given in eq. 4 is referred to the principal axes of the material to which the tensor {tilde over (d)} refers, and derives from a contraction of the indices j and k based upon the intrinsic permutation symmetry. Operatively, the elements dijk, which are also referred to as second-order nonlinear optical coefficients, are expressed as dim (with i=1, 2, 3 and m=1 . . . 6), which are obtained on the basis of the relations di1=di11, di2=di22, di3=di33, di4=di23=di32, di5=di13=di31 and di6=di12=di21.
In addition, it may be shown that, in the case where the conditions of symmetry of permutation of the indices (also known as Kleinman symmetry conditions) apply, i.e., in conditions of remoteness from possible resonances of the material and of negligible dispersion, where by dispersion is understood the dependence of the elements dijk upon the pulsation ω, the elements dijk are invariant with respect to permutations of the indices i, j, k, with the consequence that the relation dijk=dikj=dkji=dkij=djik=djki applies. There hence occurs a further reduction of the independent elements dijk, and hence a further reduction of the independent elements dim.
In addition, many materials have a second-order nonlinear optical tensor {tilde over (d)} with numerous elements dim that are substantially zero, on account of symmetries present in their own crystal lattices.
As is known, in order to determine the elements dim of the second-order nonlinear optical tensor {tilde over (d)} for a given material, the so-called “Maker-fringe method” can be applied. The Maker-fringe method, described, for example, in the article by J. Jerphagon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals”, Journal of Applied Physics, Vol. 41, no. 4, pp. 1667-1681 (1970), envisages causing an optical pump signal with pulsation ω to impinge upon a surface of a specimen of said material in such a way that the specimen will generate at output a second-harmonic optical signal with pulsation 2ω. Next, the Maker-fringe method envisages measuring the power associated to the second-harmonic optical signal and determining the elements dim on the basis of said power measurements. In greater detail, if we define as “angle of inclination” the angle at which the optical pump signal impinges with respect to the normal to the surface of the specimen, the Maker-fringe method envisages measuring the power associated to the second-harmonic optical signal as the angle of inclination varies. The variation of the angle of inclination is obtained by rotation of the specimen, for example using purposely provided actuators. The power profile thus obtained exhibits oscillations (fringes), due to the interference between two electromagnetic waves generated inside the specimen and known one as “free wave” and the other as “bound wave”. On the basis of said oscillations, it is possible to determine the elements dim.
Even though the Maker-fringe method typically envisages a so-called collinear configuration of the optical pump signal and of the second-harmonic optical signal, i.e., it envisages sending a single optical pump signal onto the specimen and observing at output from the specimen a second-harmonic optical signal having approximately the same direction of propagation as the optical pump signal, a variant of the Maker-fringe method has also been proposed. According to said variant, use is envisaged of two optical pump signals, the directions of propagation of which form an angle of mutual incidence, the direction of propagation of the second-harmonic optical signal lying substantially along the bisectrix of the angle of mutual incidence. This variant of the Maker-fringe method enables higher levels of precision to be obtained.
Even though the Maker-fringe method, and the corresponding variant, have both proven effective in enabling determination of a certain number of elements dim of optically nonlinear materials, they both present certain drawbacks.
In particular, and with reference to the Maker-fringe method, in the case where the specimen presents a large thickness (some microns), measured in the direction of propagation of the optical pump signal, the fringes are densely distributed as a function of the angle of inclination, i.e., the aforementioned power profile presents adjacent peaks corresponding to angles of inclination that are very close to one another. In said conditions, a high angular resolution is required, understood as a high precision as the angle of inclination of the optical pump signal is varied by means of actuators. In addition, in the case where, as typically occurs, the optical pump signal is formed by laser pulses having a time duration such that the spatial extent of the pulse is comparable to or shorter than the thickness of the specimen, the variation of the angle of inclination causes the optical pump signal to interact with different portions of the specimen, with consequent possible errors in the determination of the elements dim. In addition, in the case where the specimen is formed by a nanostructured thin film, i.e., by a film with two-dimensional or three-dimensional structural inhomogeneities, the rotation of the specimen causes, as the angle of inclination varies, the optical pump signal to interact with portions of specimen with markedly different characteristics.
As regards the non-collinear variant of Maker method, it enables determination of elements dim that cannot be determined using the collinear configuration, but in any case involves a rotation of the specimen, with the consequent disadvantages described above.
Two documents which relate to the determination of the second order response of centrosymmetrical materials are: “Multipolar tensor analysis of second order nonlinear optical response of surface and bulk of a glass”, F. J. Rodriguez, OPTICS EXPRESS, vol. 15, no. 14, 2007-07-09, pages 8695-8701; and “Determination of second-order susceptibility components of thin films by two-beam second-harmonic generation”, S. Cattaneo et al., OPTICS LETTERS of the Optical Society of America, vol. 28, no. 16, 2003-08-15, pages 1445-1447.